Journal of Computational Physics (J. Comput. Phys.) (20220101), 453, Paper No 110959, 22~pp. ISSN: 0021-9991 (print).eISSN: 1090-2716.
Subject
65 Numerical analysis -- 65M Partial differential equations, initial value and time-dependent initial-boundary value problems 65M50 Mesh generation and refinement
This paper presents a mesh optimization method for computational fluid dynamics. The Jacobian matrix of the semi-discrete scheme is used to identify the unstable eigenmodes based on the Lyapunov theorem of stability, where the unstable eigenmodes indicate where the mesh should be improved. A new approach is presented for moving a few vertices to stabilize the unstable eigenmodes, so that the corresponding eigenvalues are pushed to the stable side of the spectrum by using the gradients of the unstable eigenmodes with respect to vertex perturbation. Some benchmark tests show that this single iteration of the optimization is often enough to stabilize the solution, and the number of selected vertices does not grow with the problem or stencil size. Another approach is proposed to deal with the so-called opposing eigenmode cases when the stabilization may cause other modes to be unstable.